Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations (Q963013)

The authors consider the one-dimensional heat equation with selfadjoint operator. For its semidiscretized counterpart (allowing for smoothly varying grids and using a Shortley-Weller approximation), they prove uniform controllability, show a constructive way to obtain an approximation of the control, and give an estimate of the solution driven to zero for a given \(T\) which involves super-algebraic convergence in \(h\). Their results follow using discrete Carleman estimates and a discrete version (obtained here) of work by \textit{G. Lebeau} and \textit{L. Robbiano} [Commun. Partial Differ. Equations 20, No.~1--2, 335--356 (1995; Zbl 0819.35071)] concerning the loss of orthogonality of a constant (lower) portion of the set of discrete eigenfunctions when considered only on a part of the spatial domain.